Set Theory Talks

Rasmus Blanck, University of Gothenburg
Incompleteness results for arithmetically definable extensions of strong fragments of PA

In this talk, I will present generalisations of some incompleteness results along two axes: r.e. theories are replaced by Σn+1-definable ones, and the base theory is pushed down as far as it will go below PA. Such results are often easy to prove from suitably formulated generalisations of facts used in the original proofs. I will present a handful of such facts, including versions of the arithmetised completeness theorem and the Orey–Hájek characterisation, to show what additional assumptions our theories must satisfy for the results to generalise. Two salient classes of theories emerge in this context: (a) Σn-sound extensions of IΣn + exp, and (b) Πn-complete, consistent extensions of IΣn+1. Finally, I will discuss some results that fail to generalise to Σn+1-definable theories, as well as an open problem related to Woodin's theorem on the universal algorithm.

The presentation is based on the following paper: https://doi.org/10.1017/S1755020321000307

Tom Benhamou, Tel Aviv University
Intermediate Prikry-type models, quotients, and the Galvin property

We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke's result, every such intermediate model is of the form V[C] where C is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are κ+-c.c. in the generic extension and therefore do not add fresh subsets to κ+. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.

Tom Benhamou, Tel Aviv University
Intermediate Prikry-type models, quotients, and the Galvin property

We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke's result, every such intermediate model is of the form V[C] where C is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are κ+-c.c. in the generic extension and therefore do not add fresh subsets to κ+. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.

Noah Friedman-Biglin (San José State University)
Title: Regrounding the Unworldly: Pluralism and Politics in Carnap’s Philosophy of Logic

Abstract: The locus classicus of logical pluralism – that is, the view that there is more than on logic, properly so called – since the earliest days of analytic philosophy, can be found in Rudolf Carnap’s ‘principle of tolerance’. Clarifying the principle of tolerance is the focus of this first section of this paper. I will argue that the principle should be understood as widely as possible, and thus we will see that Carnap’s tolerance is a very radical view. In section two, I discuss the motivations Carnap had for his pluralism, and argue that they are based in the Vienna Circle’s “Scientific World-Conception” — a platform of philosophical commitments which set the direction for the Circle’s philosophical investigations as well as a program of social change. What emerges from this discussion is the often-ignored relationship between his logical pluralism and his political views. In short, I will argue that the radical quality of his tolerance is due to these political commitments. In section three, I examine the reasons why this connection is not very well-known. I will argue that the political situation in the United States in the aftermath of World War 2 created conditions where it was dangerous to explicitly link scholarly work and politics, and discuss the reasons that Carnap might have had for distancing himself from – or at least de-emphasizing – the political foundations of his views.

Tom Benhamou, Tel Aviv University
Intermediate Prikry-type models, quotients, and the Galvin property

We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke's result, every such intermediate model is of the form V[C] where C is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are κ+-c.c. in the generic extension and therefore do not add fresh subsets to κ+. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.

Rohit Parikh (CUNY GC).
Title: States of Knowledge

Abstract: We know from long ago that among a group of people and given a true proposition P, various states of knowledge of P are possible. The lowest is when no one knows P and the highest is when P is common knowledge. The notion of common knowledge is usually attributed to David Lewis, but it was independently discovered by Schiffer. There are indications of it also in the doctoral dissertation of Robert Nozick. Aumann in his celebrated Agreeing to Disagree paper is generally thought to be the person to introduce it into game theory. But what are the intermediate states? It was shown by Pawel Krasucki and myself that there are only countably many and they correspond to what S. C. Kleene called regular sets. But different states of knowledge can cause different group actions. If you prefer restaurant A to B and so do I, and it is common knowledge, and we want to eat together, then we are likely to both go to A. But without that knowledge we might end up in B, or one in A and one in B. This was discussed by Thomas Schelling who also popularized the notion of focal points. Do different states of knowledge always lead to different group actions? Or can there be distinct states which cannot be distinguished through action? The question seems open. It obviously arises when we try to infer the states of knowledge of animals by witnessing their actions. We will discuss the old developments as well as some more recent ideas.

Noah Friedman-Biglin (San José State University)
Title: Regrounding the Unworldly: Pluralism and Politics in Carnap’s Philosophy of Logic

Abstract: The locus classicus of logical pluralism – that is, the view that there is more than on logic, properly so called – since the earliest days of analytic philosophy, can be found in Rudolf Carnap’s ‘principle of tolerance’. Clarifying the principle of tolerance is the focus of this first section of this paper. I will argue that the principle should be understood as widely as possible, and thus we will see that Carnap’s tolerance is a very radical view. In section two, I discuss the motivations Carnap had for his pluralism, and argue that they are based in the Vienna Circle’s “Scientific World-Conception” — a platform of philosophical commitments which set the direction for the Circle’s philosophical investigations as well as a program of social change. What emerges from this discussion is the often-ignored relationship between his logical pluralism and his political views. In short, I will argue that the radical quality of his tolerance is due to these political commitments. In section three, I examine the reasons why this connection is not very well-known. I will argue that the political situation in the United States in the aftermath of World War 2 created conditions where it was dangerous to explicitly link scholarly work and politics, and discuss the reasons that Carnap might have had for distancing himself from – or at least de-emphasizing – the political foundations of his views.

Models of Peano Arithmetic (MOPA)
Monday, October 25th, 2pm
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman (vgitman@nylogic.org) for meeting id.
Fedor Pakhomov Ghent University

Rohit Parikh (CUNY GC).
Title: States of Knowledge

Abstract: We know from long ago that among a group of people and given a true proposition P, various states of knowledge of P are possible. The lowest is when no one knows P and the highest is when P is common knowledge. The notion of common knowledge is usually attributed to David Lewis, but it was independently discovered by Schiffer. There are indications of it also in the doctoral dissertation of Robert Nozick. Aumann in his celebrated Agreeing to Disagree paper is generally thought to be the person to introduce it into game theory. But what are the intermediate states? It was shown by Pawel Krasucki and myself that there are only countably many and they correspond to what S. C. Kleene called regular sets. But different states of knowledge can cause different group actions. If you prefer restaurant A to B and so do I, and it is common knowledge, and we want to eat together, then we are likely to both go to A. But without that knowledge we might end up in B, or one in A and one in B. This was discussed by Thomas Schelling who also popularized the notion of focal points. Do different states of knowledge always lead to different group actions? Or can there be distinct states which cannot be distinguished through action? The question seems open. It obviously arises when we try to infer the states of knowledge of animals by witnessing their actions. We will discuss the old developments as well as some more recent ideas.

Matteo Viale, University of Torino
Absolute model companionship, forcibility, and the continuum problem: Part II

Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory T, T∃∨∀ denotes the logical consequences of T which are boolean combinations of universal sentences. T∗ is the AMC of T if it is model complete and T∃∨∀=T∗∃∨∀. The {+,⋅,0,1}-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as ∃x(x2+1=0)∈ACF∃∨∀∖Fields∃∨∀. Any model complete theory T is the AMC of T∃∨∀. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) 2ℵ0=ℵ2 is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the ∈-theory ZFC+there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem ψ expressible as a Π2-sentence of a (very large fragment of) third order arithmetic (CH, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems ψ). Partial Morleyizations can be described as follows: let Formτ be the set of first order τ-formulae; for A⊆Formτ, τA is the expansion of τ adding atomic relation symbols Rϕ for all formulae ϕ in A and Tτ,A is the τA-theory asserting that each τ-formula ϕ(→x)∈A is logically equivalent to the corresponding atomic formula Rϕ(→x). For a τ-theory T T+Tτ,A is the partial Morleyization of T induced by A⊆Formτ.

Matteo Viale, University of Torino
Absolute model companionship, forcibility, and the continuum problem: Part II

Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory T, T∃∨∀ denotes the logical consequences of T which are boolean combinations of universal sentences. T∗ is the AMC of T if it is model complete and T∃∨∀=T∗∃∨∀. The {+,⋅,0,1}-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as ∃x(x2+1=0)∈ACF∃∨∀∖Fields∃∨∀. Any model complete theory T is the AMC of T∃∨∀. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) 2ℵ0=ℵ2 is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the ∈-theory ZFC+there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem ψ expressible as a Π2-sentence of a (very large fragment of) third order arithmetic (CH, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems ψ). Partial Morleyizations can be described as follows: let Formτ be the set of first order τ-formulae; for A⊆Formτ, τA is the expansion of τ adding atomic relation symbols Rϕ for all formulae ϕ in A and Tτ,A is the τA-theory asserting that each τ-formula ϕ(→x)∈A is logically equivalent to the corresponding atomic formula Rϕ(→x). For a τ-theory T T+Tτ,A is the partial Morleyization of T induced by A⊆Formτ.